Mesh doesn't move with armature?
you have to parent the armature to the mesh, in order for the mesh to follow the bones. 


set the 3D view in wireframe mode (Z key): it makes easier to see more than one object in the viewport
create the mesh (start with something simple, but the more vertices and subdivisions it has, the better will deform), leave it object mode
create the armature with all bones needed, leave it object mode
select the mesh and then shift-select the armature: you should then
have the mesh selected with a dark orange outline and the armature
with a lighter orange outline
press ctrl+P key and the armature parent menu shows up
for now, select armature deform -> with automatic weights
now select only the armature, and go to pose mode
select the last bone (the last in the chain) and try rotating it,
the mesh should follow. If you rotate the first bone in the chain all
other bones will follow, and so all the mesh will follow, too.
if ALL the mesh follows the single last bone, maybe the mesh
wasn't subdivided enough when you parented it. no problem:
to restart the parenting, select only the mesh (object mode) and
 press alt+P key, and choose "clear parent". then, subdivide more
 the mesh to make it more easy to deform following the single bones.
 When done, go back to point 4)


Using a multi-bone armature, each bone can be parented to specific mesh vertices, so that each bone influences the movement of the related vertices. The video you linked shows this at this point: https://youtu.be/n0VspDUOErE?t=421 

The relation between a bone and mesh vertices is not exclusive of that bone, though. Other bones could influence at least part of the same vertices, if you decide so. Each bone can be set to influence a set of vertices with a variable degree of "weight". When more than one bone moves, all the influenced vertices are moved following the overall weights attributed to all parented bones of the armature.

This is a process usually known as "skinning", you can learn how here: http://www.blender.org/manual/rigging/skinning/obdata.html